A364344 Number of endofunctions on [n] such that the number of elements that are mapped to i is a multiple or a divisor of i.

1, 1, 4, 20, 177, 1462, 21919, 254802, 4816788, 82401465, 1929926410, 35256890748, 1152938630784, 24977973856643, 823036511854847, 24332827884557037, 954801492779273665, 27023410818058291822, 1309814517293654535339, 41375530521928893861920

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Example

a(0) = 1: ().
a(1) = 1: (1).
a(2) = 2: (11), (12), (21), (22).
a(3) = 20 (111), (112), (113), (121), (122), (123), (131), (132), (211), (212), (213), (221), (223), (231), (232), (311), (312), (321), (322), (333).

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)+add(
     `if`(d>n or d=i, 0, b(n-d, i-1)*binomial(n, d)),
          d=numtheory[divisors](i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..19);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i - 1]* Binomial[n, i*j], {j, 0, n/i}]+Sum[If[d>n || d == i, 0, b[n - d, i - 1]* Binomial[n, d]], {d, Divisors[i]}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Oct 27 2023, after Alois P. Heinz *)

Crossrefs

Cf. A000312, A178682, A334370, A364327, A364328.

Sequence in context: A185672 A210438 A054474 * A368455 A213144 A215873

Adjacent sequences: A364341 A364342 A364343 * A364345 A364346 A364347

Keyword

nonn

Author

Alois P. Heinz, Jul 19 2023

Status

approved